The quotient of limits rule is a limit operation in calculus and it expresses that the limit of quotient of two functions is equal to the quotient of their limits. Now, let’s learn how to prove the limits division rule in mathematics.
Firstly, let’s consider two functions, whereas every function is defined in a variable $x$, and the functions are simply written as $f(x)$ and $g(x)$ in mathematics. Let’s assume that the function $f(x)$ is divided by function $g(x)$, and the division of $f(x)$ by $g(x)$ is written as $f(x) \div g(x)$ in mathematics.
The limit of division of functions as its variable $x$ approaches $a$, is written as follows.
$\displaystyle \large \lim_{x \,\to\, a}{\normalsize \big(f(x) \div g(x)\big)}$
The limit of the quotient of $f(x)$ divided by $g(x)$ as the value of $x$ tends to $a$, is written mathematically as follows.
$=\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \dfrac{f(x)}{g(x)}}$
Now, let’s learn the process of proving the limits quotient rule by finding the limit of the quotient.
Let’s use the limits by direct substitution method to find the limit. Now, substitute $x$ equals to $a$ in the rational function to find the limit of quotient of functions as the value of $x$ tends to $a$.
$\implies$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \dfrac{f(x)}{g(x)}}$ $\,=\,$ $\dfrac{f(a)}{g(a)}$
Now, let’s find the limit of both functions individually as the value of $x$ approaches to $a$ by direct substitution method.
Let’s substitute $x = a$ in the function $f(x)$ to find its limit as $x$ approaches to $a$.
$\implies$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}$ $\,=\,$ $f(a)$
The limit of function $f(x)$ as the value of $x$ approaches to $a$ is $f(a)$. So, it can be written as follows.
$\,\,\,\,\,\,\therefore\,\,\,$ $f(a)$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}$
Similarly, let’s substitute $x = a$ in function $g(x)$ to find its limit as the value of $x$ tends to $a$.
$\implies$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g(x)}$ $\,=\,$ $g(a)$
The limit of function $g(x)$ as the value of $x$ tends to $a$ is $g(a)$. So, it can also be written as follows.
$\,\,\,\,\,\,\therefore\,\,\,$ $g(a)$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g(x)}$
Now, let’s find the mathematical relationship between the functions for proving the division law for limits.
We evaluated above that
$\implies$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \dfrac{f(x)}{g(x)}}$ $\,=\,$ $\dfrac{f(a)}{g(a)}$
We have also evaluated above that
$(1).\,\,$ $f(a)$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}$
$(2).\,\,$ $g(a)$ $\,=\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize g(x)}$
Now, substitute the values of $f(a)$ and $g(a)$ in the above equation.
$\,\,\,\,\,\,\therefore\,\,\,$ $\displaystyle \large \lim_{x \,\to\, a}{\normalsize \dfrac{f(x)}{g(x)}}$ $\,=\,$ $\dfrac{\displaystyle \large \lim_{x \,\to\, a}{\normalsize f(x)}}{\displaystyle \large \lim_{x \,\to\, a}{\normalsize g(x)}}$
Therefore, it has proved that the limit of a quotient is equal to the quotient of their limits. It is called the quotient rule of limits and it is also called the division rule of limits.
The multiplication rule of limits with proof to learn how to find the limit of a product of functions.
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